Calculator



' Dec. 26, 1950 s.- SKOLN4|K 2,535,374

CALCULATOR Filed June 15, 1946 4 Sheets-Sheet 1 v INVENTOR Samue/fiko/n/k ATTORN EY S. SKOLNIK Dec. 26, 1950 CALCULATOR 4 Sheets-Shegt 2Filed June 15. 1946 J Ig 4 12a INVEN TO K Samue/ Ska/0M ATTORN EY Dec.26, 1950 s. SKOLNIK 2,535,374

CALCULATOR Filed June 15, 1946 4 Sheets-Sheet 3 lg. 121 120 82 so 1 w.74 K 910LL134BI5 1' 1'7 IIB\L J .58 fig. 6

INVENTOK k 1 Samua/ s/m/m/f ATTORNEY Dec. 26, 1950 s. SKOLNK 2,535,374

\ CALCULATOR Filed June 15, 1946 4 Sheets-Sheet 4 n3. 7 .NVENTOR Samue/SAo/n/k BY W 3%.

ATTORNEY Patented Dec. 26, 1950 UNITED STATES PATENT OFFICE CALCULATORSamuel Skolnik, Les Angeles, Calif.

Application June 15, 1946, Serial No. 676,871

6 Claims. 1

This invention re ates to a calculating device.

In an ordinary Mannheim-type slide rule, the C and D scales usually usedfor multiplication and division carry marks subdivided from 1 to 10. Thenumbers correspond, in these scales, to the lengths of the logarithms ofthe numbers.

Physical addition and subtraction of scale lengths thus provides amethod of multiplication and division. It is not practical to use ascale length of more than about a foot. Therefore, the degree ofaccuracy, depending often on interpolation, does not exceed threesignificant figures.

It is one of the objects of this invention to make it possible to uselong scales in a relatively compact manner, and thereby to obtainaccuracy corresponding to at least four significant figures.

In order to accomplish this result, sections of a logarithmic scale,disposed in a convenient manner, as on a chart, are provided, and which,with a pointer, can be used to multiply or divide. In this arrangementthere is no need to provide relatively movable scales. instead, only apointer joining two of the factors is required.

In the operations involving these sectional scales, an index line may beused, connecting two points on the scale and corresponding to the fac-'tors of the computation. At times, such a line may make a small anglewith the scale, rendering reading the scal s diiiicult and inaccurate.It is another object of this invention to provide a substitute for sucha line in which the angle of the line with respect to the scale isalways sufiiciently great to preclude any material error of observation;for example, the minimum angle of intersection can be readily made aslarge as 45.

The provision of sectional scales utilized in this manner is useful forcalculations other than multiplication or division. As anoth r example,sectional scales, appro riately subdi ided and indexed, may be us d tosolve problems involving the square root of the sum of the squares ofnumbers. Thus given two of the three sides of a right triangle, thethird side can be expeditiously obtained.

It is, accordingly, another obiect of this invention to provide a devicethat can be used to solve for any of the three numbers in the relationThis invention possesses many other advantages, and has other objectswhich may be made more clearly apparent from a consideration of severalembodiments of the invention. For this purpose there are shown a fewforms in the drawings accompanying and forming part of the presentspecification. These forms will now be described in detail. illus t g eg ne a pr nciples of the invention; but it is to be understood that thisdetailed description is not to be taken in a limiting sense, since thescope of the invention is best defined by the appended claims.

Referring to the drawings:

Figure 1 shows a chart incorporating the invention, utilized formultiplication and division;

Figs. 2 and 3 are diagrams illustrating the principles of the invention;

Fig. 4 is a view of a pointer or index adapted to be used with the chartof Fig. 1;

Figs. 5 and 6 are fragmentary enlarged views, illustrating the manner inwhich the pointer of Fig. 4 is utilized in connection with the chart ofFig. 1; I

Fig. 7 shows a chart adapted to be used to obtain square roots of thesums of squares of two numbers, or the square roots of the difference ofthe squares of two numbers, many of the scale divisions being omitted;

Fi s. 8 and 9 are views, similar to Figs. 5 and 6, illustrating themanner in which the pointer of Fig. 4 may be used with the chart of Fig.'7; and

Fig. 10 is a fragmentary view, illustrating another form of pointer usedwhere the numbers involved lie close together on a single scale section.

The chart A of Fig. l is subdivided logarithmically. It includes twentyuniformly spaced scale section's l to 20 inclusive. These sections areall of uniform length, the left-hand ends of the scale falling upon acommon straight line 2| In the present instance, the twenty sectionsform a scale corresponding to logarithms of the numbers B, a pearing atthe scale divisions above the scale sections, the numbers corresponding,to the logarithms of one to ten, or from to 1000. In order tofacilitate computations, the zero point C of the scale, corresponding tothe left-hand end of scale section I, is marked with the number 100.Accordingly, t e end of the scale, corresponding to the right-hand end Eof section 20, corresponds to the logarithm of ten or 1000. The numbersabove the scale sections carry indicia using three significant figures.In this way, no numbers (except the number 1000 corresponding to pointE) have more than three digits, whereby the scale numbers can be morereadily identified. Since the decimal point in operations of mu1tipli-'cation and division, when using logarithmic scales, are quite easilydetermined, and only the digits are of significance, this omission oraddition of zeros in no wise affects the use of the scales.

In this mode of division, the numbers B, marked on the scale above thesections l to 20 inclusive, correspond to the lengths of the logarithmsof the numbers. of the number 112 extends almost to the righthand end ofscale section I, the space between thenumbers decreasing as the scaleprogresses,

Thus, the logarithm The scale sections I to 2i! inclusive also carryother graduations F4 marked ofr" below each scale section I to 28inclusive. Th se lower numbers correspond to the square of the. numbersB appearing immediately above the. corresponding F graduations.

Thus, for example, in the scale section 1 the.

number 200 appears near the left-hand end of the scale line. Immediatelyunder it is the mun"- ber 400, corresponding to the square of.200, withtwo zeros omitted.

It is obvious that. in this manner: the twentyscale sections l to 28inclusive correspond to a complete C or D scale of a conventionalMannheim slide rule. The overall length of the completescale is muchgreater than could be obtained inla conventional slide rule; all of thesect ons" being arranged compactly: on. a: sin le chart AThescalesections l to'2 l 'are'ma'rked"in heavy lines on the chart.andcanbe used in amanner to be hereinafter described for multiplicationor division.

In order to obt'aiuthe results of the multipl cation or division of twonumbers-by the aid ofthi's chart; use may be made of a pointer" or indexwhich may be in" the form ofa straight'iline;

Thus, for example; as shown in Fig; 2; the

two numbers to be multiplied can be represented! by the points 2 2 and23' on the B sealer The point 22 may appear. for example, on scale-section' 3 and the point 23 on scale section 1;: and

these scale= sections" are separated by an even.

number of spaces; four in this instance. straight line 25, joining thee" points: intersects the scale section 5 at the point25. the s alesectionl5 being equidistantfrom the scale sections Sand 11 It canreadilybe'sho n that the oint 25 corre sponds to a'. scale division onthe scale F" below these' tion 5. representingthe product of thetwo-cm'antitiescorre pondinc tn the points 22 and 2-3i. This can beshown as follows: If we" desi thate the: num r on th scal F'at point 3as m; and the'number on the scale F at'point'2'3 as" n22 and thenumbercorresponding to point 25 A cord ncl th s ua e of n read. on scaleF' at'point 25' isthercorr tr sult:

Obviously th srelati nshi holds'for'all nurn' bers'that are oine b"lines ect'e at'point 5. s ch fore ampl'e, as indi'atd bypoints 2? and 28f Fig; 21

Theinverseproce s ofdivi ion can'be carried out bysetting the lineZ'i'sothat it passes thI'OllTh the product 2 on scale-F 5 and also so thatitpasses'throne'h the-divider such as thenumber" corresponding to scaledivision 23. The answer isreadoif'atpoint 22 on the upper scaledivisions.

It'mav occur'thatthe two factors'do-not lie on section -lines that arespa ed by an even n mb r of spaces: Such a conditionis repres nted inFig. 31 where there are but-three spaces between the topscale section 3and the lower scale section 6 The two-numberstobe"mulitplied; corretothepoints 35 and: 31;

spending to the points 38 and 3 I, are read on the upper scale divisionsB of these sections. In this case the center point 32 of line 29 islocated between the scale sections 4 and 5. In order to be able to readthis value; corresponding to. the bisecting point 32, another set ofscale sections, shown in light lines, is provided intermediate the mainscale sections 1 to 20. These scale lines aremarked by referencecharacters 34 to 52 inclusive on Fig. 1. Of course, the two sets ofscale sections l to 20, and 34 to 52 inclusive, can be formed by linesof contrasting colors, or in some other way readily to distinguish them.In this instance the intermediate scale section lines are shown as muchlighter in weight than themain' section lines.

Theiscale G onsections' 34 to 52 corresponds to' scale. F'in thefollowing manner. Thel-18ft'- hand end 53 of the first intermediatescalev 34" corresponds to the center of the scale F immediately-aboveit;and, from this point 53, scalei Gfollows the same'as the'scaleF.

It may' readily be. shown that the correct. product can be read-on thescale division G of 1 the intermediate" scale in" the event thecenterofthe line 29 (Fig. 3') falls'on" one ofI'theTinte'rmediate scalesections, such as 3?. The scale: division on scale G at the point 32gives the: proper product for: the numbers" corresponding Thiscan"readily' be: demonstrated in a manner similar. to" that set: forth with:respect to'Fig; 2.

The: process of division for: cases such as in Fig; 3 is accomplishedasbefore. The: line 29 i is caused to pass through point 32.. Thedivider" maycorrespondto-thepoint 3001 point 3 I. The resultcan be readat point 3lor 30- upon the upper-scale divisions 6' or 3;

The scales can obviously also be" used: for ob'-- taining the-product ofthe square. roots of'numbers, or-the products of the square root-ofone"number and another number; or the quotient: of thesquare of anumber-and!another'numbeml or the quotient of the squares of" twonumbers;

It is: also possible, by the aid of additional indicia; to obtain reults" of formulae generally that: utilize'two factors tabs-multiplied;as this: involves merely' a matter of indexing;

In: the event: line-2'6" or-29imakes a'smallang1es with the' scale sctions, the accuracy; of' reading thescale divisions may be." reduced.

It is possible to ensure that the scale divisions; can: leed termin dmore accurately. by providing a point"r or'indexin which the angle ofinter.- "ectionwith'the scale line isalivays considerable; For thispurpose. a pointer or index J (Figs; 1' ard 4) is provided, preferablyformed" of thin transparent material" such as C lluloid. ortrac ingcloth; It carries a series of curves 53 to'65' inclu-ive;

Th point r J has an index point 66 soar-- ranged that a"y line, suchas'fi' l drawnfrom the poirt 56 will intersect the curves 5'6 to65inclufiive at a conveniently large angle, such as near 45.

Assumin t at a line 61 is-drawn between point 66 so as to end on theoutermost curve 56, the curves are so: arranged that. the segment 66-68of linet-I isbisected' at the point 59, correspondin; totheintersection'ofthe'line-fi'l onthercurve 58;.the. n xt but on to:curve 56. Similarly, the segment (Stii) is; bisected at the point: 1:!0111 curve 53., the next butoneto' curve 51.; onwhich point. 1 llfall's.

gi n other words; any segment beginningat point L 6 tan a where r and 0are the polar coordinates of any points on the curve, C is a constantfor each curve, a is the angle of intersection of the curve with theline 51, and e is the natural base of logarithms. The constant C ischosen for each successive curve 56 to 65 so as to provide the bisectingrelation hereinabove mentioned. When a. is 45, as indicated herein, theform of the equation is simplified, for then tan (1 equals 1. Further.each of the curved portions 55 to 65 are long enough so that theshortest radius of each of the curved portions is at least as long asthe longest radius of the next smaller curve. In this way, it is assuredthat all lengths can be measured continuously within the limits of thepointer.

The use of the index J is illustrated in Fig. 5. In this case the number112, a pearing on scale section I, is to be multiplied with the number160 ap earing on scale section 5. The point lifi is plac d on the scaledivision I I2. and the ind x J is then moved about division 2 until oneof the curves 56 to 65 inclusive intersects the scale di isioncorresponding to 60 on section 5. h n, since there are an even n mber ofspaces between the scale section I and the scale section 5, the answeris read off by the aid of curve 58 (the second from curve 56) where itint rs cts the scale section 3: and t is an wer is read oil. on the lowr scale divisions F of scale section 3.

So long as th radius vector line 55 mak s but a small angl with the scallin s I to 20. th in tersections of the curves with these scale lines isin the nei hborhood of 45,and an accurate readin can be obtained. Wherethe an-le between the radius vector, such as 55. and the scale linesapproaches 45, it is advisable to rev rse the ind x member J in ord r toobtain a sufliciently larger an le of int r ection between the. scalelines and the curves 55 to 65. Such a situation is actually present inFig. 5; and Fig. 6 illustrates a manner of use of the member J to obtaina larger angle of intersection. To ensure accuracy and reduce parallax,the member J can carry the curves and other marks on both sides thereof.

In the instance shown in Fig. 6, the number 125, appearin on scalesection 2, is to be multipli d with the number 260 appearing on scalesection 9. Since there are an odd numb r of spaces between scalesections 2 and 9. an auxiliary scale section 38, shown in light lines,is used to read off the answer on scale G at the intersection 13 ofcurve 59 with section 38.

Due to the fact that the length of the complete scale in the form ofFig. 1 is quite large, the answer can be read off easily to four figureplaces.

The chart M of Fig. '7 is provided for the solution of problemsinvolving three numbers of the form In this instance, there are againtwenty main scale sections, 14 to 93 inclusive. In this instance, thescale section subdivisions K, above the scale lines, correspond with thelengths of the squares of the numbers. Thus, the distance from theleft-hand end 94 of the first scale section I4 to any number, such as17, corresponds to a distance representing the square of the number.Here, again, the numbers from 1 to are accommodated by the twenty scalesections 14 to 93.

i The lower scale divisions L here correspond to numbers equal to thesquare root of 2 times the numbers appearing on the upper scaledivisions K.

Auxiliary scale sections 91 to H5 inclusive are evenly spaced betweenthe main scale sections. They carry similar subdivisions, but, asbefore, the left-hand ends of each of these scale sections correspond tothe scale division at the center of the main scale section immediatelyabove it.

It may be readily demonstrated that the square root of the sum of thesquares of two numbers can be obtained by the use of the index member Jas before. Thus, for example, in Fig. 8 the square root of the sum ofthe squares of the numbers 30 and 66 is desired. The number 30 appearson the main scale line 15, and the number 66 annears on the main scaleline 82. The point 36 of the index member J is placed on the num ral 30and one of the curves, such as 58, falls on the scale divisioncorresponding to 66 on scale line 82. Since there is an odd number ofspaces between scale lines 15 and 82, the answer can be read on thescale P of the auxiliary scale section I l. corresponding to the pointH6 where curve 65 intersects section NH. H re. as in the case of Fig. 6,the index is inverted to provide int rs ctions makin large anglesbetween the curves and the scale lines.

In Fl. 9, the application of the chart M is indicat d when an evennumber of spaces separat s the scal s ctions on which the numbers appar. Thus, in this case, the square root of the "um of the squares of 30and 40 is required. In this instance, t e pointer or index member J 'isplac d so hat the point it falls on the number an on scale line 15. Oneof the curves. such as 5'! is then caus d to int rsect a scale line at adivision I H corresponding to the number 40. This a pears on scalesection 17. The answer is read on the int rm diate scale section 16, thelower scale subdivision L thereon corresponding to the number 50. Here,since the radius vector 55 to oint 49 of curve 5". makes a small anglewith the section lines, the index J is used in uninvert d position.

The reverse process of finding the square root of the differ nces ofsquares of two numbers may obviously also be r adily accomplished bythis m thod. The method is quite general.

.In the event the two numbers to be treated fall upon t e same scalesection, the index J can be used to bisect the distance between thesetwo values.

;Furthern ore, in the event the two numbers happen to fall close toether on the same scale section a different kind of pointer may be used.Thus, in Fig. 10, scale section 14 of the chart M of Fig. 7 is shown. Inthis case it is desired to find the square root of the sum of thesquares of 8 and 10. The member J carries an isosceles triangle H26 inwhich is also drawn the altitude I21. This line iZI obviously bisectsany line drawn parallel to the base of the triangle.

Accordingly. as shown in Fig; 10, this isosceles triangle may be placedso that its two legs fall respectively on the scale sectionscorresponding to the numbers 8 and 10, the base of the triangle beinheld parallel to the scale section 14. The

altltudeiine l 2| thenbisects the distance betweend 8 and- 10,uand -the"answer can"be read on the lower scale L: This-answeris-128.

Inithe forms of thechartsshown in Figs." 1 and 7, the numbers carried bythe lower subdivisions E-and' L areresp'ectively related to the numberscarried by the upper subdivisions B and K, re

spectively; by a constant exponent; e.--g:, the exponent 527 in the form.of Fig. 1, since the lowernumbers are squaresof the uppernumbers.

The inventorclaims:

1. Anpointer formedmof translucent material,

havingjal-seri'esbf' spaced curves thereon; as well as' an indexpoint,*said curves being such that anystraight line drawn from i thepoint inter sects I all of the curves; and is divided proportion--atelyr' by said: intersections, the' length of i any straight line'toany intersection on any-curve beingr"bisected"'by that'curvewhich istwo spaces 1 nearer the point "than: said intersection;

2." A pointerformed of translucent material; having "a series-of spacedcurves thereon, as'well" as an index point; said'curves being suchthatany"straight line drawn from the point -inter= sects all of thecurves, and is divided'proportlom atelyrby said intersections, saidcurves corre* spending in polar coordinates to the form where-1r and .0are the :polar coordinates of 1 any.

intersection; wis the angle made by the curve with the straight line; Cis a-constant; and c is the base of natural logarithms.

35- A piointer formed of translucent -material,

having a' series" of spaced curves thereon, as well" as an index point,"said curves being such that I any? straight line drawn from the pointinter-- sects all of the curves; and'is-divided proportionately by saidintersections, the length-of any stralght line to any intersection onanycurve beingjbisected by that-curve which is two spaces nearer the pointthansaid intersection; said curves corresponding in polar coordinatestosthei form:

ra, v tan a wherer' and are the polar coordinates of any intersection;0. isthe angle made'by' the'curve there being two sets of scaledivisions for each line: the first of said scale divisions havingnumbered spacings in a definite ascending series, the

successive scale sections forming a" continuous uninterrupted scale; theother of said scale divisions having numbered spacings related tothe'first scale divisions by a constant exponent: there being another set"of sectionai scale'lines equally" shaced'b'etween the lines of the Jfirst setand similarlysubdivided; the beginning point of the"- firstscale section of said I second 'set' correspondin'g toithecenter of thefirst sectional 'linerof the first set, and indexediiniaccordance withsaidv other: scale? divisions; 4 and a. pointer member Haring; a point-forigin a and spaced; linesfor 8t dividing; a "line wsegment into equal Ipar-ts; l therorigin 7 and the pointer IinesbeingJ 'adapted: to 'roverlie selected scale divisions.

5.--A chart having apluralityj of sectional, uni"- formlyspaced'parallel-scale lines of equal lengths;

the ends of the lines falling on:a common-straight: line perpendicularto .the parallel scale lines; there being two sets-of scale divisionsfor each line, the first of said scale divisions having numberedspacings corresponding in length from the beginning of the first line tothe ilogarithmcf the number; the successivescalesections formingsacontinuous uninterruptedlogarithmic scale; the other of 'said scaledivisions having. numbered? logarithmic scale lines, the numbers thereonx corresponding to thevsquares .of-the numbers 0152 the first scaledivisions; there beinganotherset. of sectional scale lines equallyspaced between the lines of the'first set and subdivided and numiberedin accordance with said other scale di'visions, the beginning. point ofthe first scale' section-of said second set corresponding to the centerof the firstsectional. line of the firstlset; and: a pointermemberhaving a point .of origin and a series of spacedcurves forming aconstant angle.

with any line drawn from the origin, the alternate curves-dividinga-line from the origininto equal parts, the origin and the pointer linesbeing. adapted to overlie selected scaledivisions.

6; A chart having aflplurality of sectional," uni formly spaced parallelscale lines of equal length's the ends of the lines falling on a commonstraight r line. perpendicular to-the parallel scale h lines;therelbeing two sets of scale divisions. for each line, the first ofsaid scale-divisionshaving num bered spacings corresponding in lengthfrom the beginning of-thefirstlihe to thesquare of the number; thesuccessive scalesections formingra continuous uninterrupted scale; theother of saidscale divisions having numbered scale lines, the numbersthereon corresponding to the square root-of two times the numbers on thefirst-scale divisions; and apointer member having a-point of origin and-a series of spaced curves .forming, a constant angle with any linedrawn from the origin, the alternate curves dividing a linefromthe'origin into equal parts, the origin and the" pointer lines beingadapted tooverlie selected scale divisions.

SAMUEL SKOLNIK.

REFERENCES"CITED Thefollowing references are of -recordin-thefile ofthis patent:

UNITED STATES PATENTS Number" Name Date 802,344 Wilkes Oct.-17,'l9051384, 176 Haimes Feb. '19, 1926: 1,610,706 Ragot et 'al. Dec 14, 19261,632,505 Ritow -s June 14, 1927 $075,854 Karnes s Apr. 6, 19372,232,319 Gay Feb? 18,1941

FOREIGN PATENTS Number H Country Date 3;685" Great Britain Feb. 14, 1907OTHER REFERENCES Pages 6, 45, 47 and 132 of Graphical and Me' "chanicalComputation? by Joseph Lipka, published by John Wiley and Sons, Inc., N.Y., 1918.

